Critical-point slope of barotropic spherical accretion ($C=\rho d(c_s^2)/d\rho$)

At a regular sonic point, differentiating the vanishing coefficient and numerator yields $(2+C/c_s^2)(u^{\prime }{)}^2+4C{u}^{\prime }/(ru)+(4C+2c_s^2)/r^2+{\Phi }^{\prime \prime }=0$. A real slope compatible with the desired branch is needed in addition to the sonic equalities. For constant sound speed this reduces to $(u^{\prime }{)}^2=-c_s^2/r^2-{\Phi }^{\prime \prime }/2$.